What is the correct way of writing a vector element?

Solution 1:

In my opinion, keeping the vector bold is more correct, because you are applying the “extract the $i$-th component” function to the vector.

However, in my experience it is far more common to use the regular font when referring to the components.

Solution 2:

I prefer the first. I think of starting with an indexed set of scalars $\{x_i\}$ (not bold, italicized, as normal) for $i \in \{1,\ldots,n\}$. Then from those scalars I build $\mathrm{\mathbf{x}}=[x_1,\ldots,x_n]^{\top}$, an object with a different data type (vector) which uses math roman and bold to indicate its different type.